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Theorem isoval 13982
Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
isoval.n  |-  I  =  (  Iso  `  C
)
Assertion
Ref Expression
isoval  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )

Proof of Theorem isoval
Dummy variables  x  c  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
2 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5720 . . . . . . . 8  |-  ( c  =  C  ->  (Inv `  c )  =  (Inv
`  C ) )
4 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
53, 4syl6eqr 2485 . . . . . . 7  |-  ( c  =  C  ->  (Inv `  c )  =  N )
65coeq2d 5027 . . . . . 6  |-  ( c  =  C  ->  (
( z  e.  _V  |->  dom  z )  o.  (Inv `  c ) )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
7 df-iso 13967 . . . . . 6  |-  Iso  =  ( c  e.  Cat  |->  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  c )
) )
8 funmpt 5481 . . . . . . 7  |-  Fun  (
z  e.  _V  |->  dom  z )
9 fvex 5734 . . . . . . . 8  |-  (Inv `  C )  e.  _V
104, 9eqeltri 2505 . . . . . . 7  |-  N  e. 
_V
11 cofunexg 5951 . . . . . . 7  |-  ( ( Fun  ( z  e. 
_V  |->  dom  z )  /\  N  e.  _V )  ->  ( ( z  e.  _V  |->  dom  z
)  o.  N )  e.  _V )
128, 10, 11mp2an 654 . . . . . 6  |-  ( ( z  e.  _V  |->  dom  z )  o.  N
)  e.  _V
136, 7, 12fvmpt 5798 . . . . 5  |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
142, 13syl 16 . . . 4  |-  ( ph  ->  (  Iso  `  C
)  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
151, 14syl5eq 2479 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
1615oveqd 6090 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
17 eqid 2435 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x (Sect `  C ) y )  i^i  `' ( y (Sect `  C )
x ) ) )
18 ovex 6098 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1918inex1 4336 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
2017, 19fnmpt2i 6412 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B )
21 invfval.b . . . . . . 7  |-  B  =  ( Base `  C
)
22 invfval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
23 invfval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
24 eqid 2435 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
2521, 4, 2, 22, 23, 24invffval 13975 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
2625fneq1d 5528 . . . . 5  |-  ( ph  ->  ( N  Fn  ( B  X.  B )  <->  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C )
y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B ) ) )
2720, 26mpbiri 225 . . . 4  |-  ( ph  ->  N  Fn  ( B  X.  B ) )
28 opelxpi 4902 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2922, 23, 28syl2anc 643 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
30 fvco2 5790 . . . 4  |-  ( ( N  Fn  ( B  X.  B )  /\  <. X ,  Y >.  e.  ( B  X.  B
) )  ->  (
( ( z  e. 
_V  |->  dom  z )  o.  N ) `  <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) ) )
3127, 29, 30syl2anc 643 . . 3  |-  ( ph  ->  ( ( ( z  e.  _V  |->  dom  z
)  o.  N ) `
 <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z ) `  ( N `  <. X ,  Y >. ) ) )
32 df-ov 6076 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
33 ovex 6098 . . . . 5  |-  ( X N Y )  e. 
_V
34 dmeq 5062 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
35 eqid 2435 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
3633dmex 5124 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
3734, 35, 36fvmpt 5798 . . . . 5  |-  ( ( X N Y )  e.  _V  ->  (
( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y ) )
3833, 37ax-mp 8 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y )
39 df-ov 6076 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
4039fveq2i 5723 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
4138, 40eqtr3i 2457 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
4231, 32, 413eqtr4g 2492 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
4316, 42eqtrd 2467 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   <.cop 3809    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870    o. ccom 4874   Fun wfun 5440    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Basecbs 13461   Catccat 13881  Sectcsect 13962  Invcinv 13963    Iso ciso 13964
This theorem is referenced by:  inviso1  13983  invf  13985  invco  13988  isohom  13989  oppciso  13994  funciso  14063  ffthiso  14118  fuciso  14164  setciso  14238  catciso  14254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-inv 13966  df-iso 13967
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